Re: [Phys-l] Basic statistics

[Jack Uretsky <jlu@hep.anl.gov>]

Hi Tim-
	I misremembered, and therefore mis-stated, what it is that the 
Chi-square distribution does not do.  You correctly state that it becomes 
Gaussian in the large sample-size limit.  What it does not do is become 
n(0,1) (0-mean, unit variance) as some other distributions do.  The 
zero-mean part is, of course, trivial because it just invloves subtraction 
of the the distribution means.
	The correct statement of the CLT is that the means of samples 
taken from finite variance distributions tend to n(0,1) in the large 
sample-size limit.  The are distributions for which the CLT fails.  See 
Feller, 2d Ed., Vol. II (not an easy read).
	I think we're not far apart.
				Regards,
					Jack


On Sat, 11 Nov 2006, Folkerts, Timothy J wrote:

> Jack
>
> You & I seem to be talking past each other.  Let's see if I can make my point more clearly and then see if we can reach common ground.
>
>
> > When I said "carefully stated" I was trying to exclude puns.
> > Tim  makes a pun on the phrase "applies to."
>
> While I like puns as much - usually more ;-) - than the next guy, I'm not sure what I said that could be construed as a pun.
>
> Your basic statement of the CLT was
> "The texts tell us, as I understand them, that the means of vary
> large samples taken from finite variance distributions are normally
> distributed about a central value.  I do not know whether there is a proof
> that relaxes the limiting conditions."
>
> That sounds like a pretty decent summary.
>
>
> > While his statement is correct, it doesn't contradict mine,
> > namely, the chi-square distribution does not become normal
> > in the large N limit.
>
> and
>
> > In the case of the chi-square distribution it is easy to demonstrate
> > by example with MathCad or a similar program that the large N limit
> > is not Gaussian.
>
> I still disagree.  I ran a simulation using a chi square with 2 degrees of freedom (which looks far from normal).  I added 100 (i.e. a large N) results from this chi square distribution.  I repeated this process 1000 times.  The distribution of these 1000 points looks normally distributed.  It also passes the standard Anderson-Darling test for normality.  (Or more precisely, it doesn't fail the test.)
>
> And actually, I will admit I WAS wrong on one point.  The chi-square distribution does approach a normal distribuion for large degrees offreedom.  It doesn't look very close for small degress of fredom, but it does eventually start to look like the normal distribution.
>
> So whether N means "degrees of freedom" or N is a large sample as intended for the CLT, either one approaches normality for large N.
>
>
> Tim F

--
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